Exam 17: Inference About a Population Proportion

A study examined the effectiveness of Botox injections in the corrugator supercilii muscles for the treatment of chronic migraines. Of the 29 migraine-prone patients participating in the study, 16 reported complete migraine elimination. Using technology and a plus four 90% confidence interval for the population proportion of patients with complete migraine remission following such Botox injections, what is the margin of error? Assume the necessary assumptions for this confidence interval are valid.

A large fishing farm with thousands of fish has been treating its fish to stop a spreading fungal infection. A simple random sample of 50 fish is taken to determine the proportion p that is infected in this population. A careful examination determines that 6 of the fish sampled are infected. The infection would be contained if fewer than 10% of the fish were infected. Using appropriate technology, what is the numerical value of the test statistic for the correct set of hypotheses?

People can develop allergic reactions to sulfites, a family of chemical compounds often used for commercial food preservation. Researchers examined a sample of 21 patients with hypertension who had sought treatment regarding some allergic reaction. The published findings state that 9 of them were sensitive to sulfites. How large a sample n would you need to estimate p with margin of error 0.05 with 95% confidence? Use the guess p = 0.50 as the value for p*.

A study examined the effectiveness of Botox injections in the corrugator supercilii muscles for the treatment of chronic migraines. Of the 29 migraine-prone patients participating in the study, 16 reported complete migraine elimination. How large a sample n would you need to estimate p with margin of error 0.02 with 90% confidence? Use the guess p* = 0.50 as the value for p.

A survey of food allergies in a simple random sample of 184 young children found that 13 had some kind of food allergy. What is the margin of error for a plus four 95% confidence interval for the true population proportion of young children who have a food allergy?

A study examined a simple random sample of smokers who tried to quit smoking by using either a nicotine patch or e-cigarettes. Here is a quote from a news article referencing the study: "Quitting is hard. The number of people who quit (within 6 months) was low overall-just 38 of the 584 smokers given the e-cigarette or the patch." Using technology, what is the plus four sample estimate of the proportion of smokers who quit within 6 months using either e-cigarettes or the nicotine patch?

A survey of food allergies in a simple random sample of 184 young children found that 13 had some kind of food allergy. What is the plus four sample estimate of the proportion of young children who have a food allergy for this sample?

A simple random sample of 900 13- to 17-year-olds found that 411 had responded better to a new drug therapy for autism. Let p be the proportion of all teens in this age range who respond better. Suppose you wanted to see whether the majority of teens in this age range respond better. To answer this question, you test the following hypotheses: H₀: p = 0.50, H_a: p > 0.50 Using appropriate technology, what is the P-value of your test?

An inspector inspects a shipment of medications to determine the efficacy in terms of the proportion p in the shipment that failed to retain full potency after 60 days of production. Unless there is clear evidence that this proportion is less than 0.05, she will reject the shipment. To reach a decision, she will test the following hypotheses using the large-sample test for a population proportion: H₀: p = 0.05, H_a: p < 0.05 To do so, she selects a simple random sample of 200 pills. Suppose that 8 of the pills have failed to retain their full potency. Using appropriate technology, what is a 95% plus four confidence interval for the true proportion of pills in the shipment that have failed to retain full potency based on these data?

Before performing a hypothesis test for a proportion H₀: p = p₀, what must we demonstrate?

A study examined a simple random sample of smokers who tried to quit smoking by using either a nicotine patch or e-cigarettes. Here is a quote from a news article referencing the study: "Quitting is hard. The number of people who quit (within 6 months) was low overall-just 38 of the 584 smokers given the e-cigarette or the patch." Are the conditions for this inference procedure met?

People can develop allergic reactions to sulfites, a family of chemical compounds often used for commercial food preservation. Researchers examined a sample of 21 patients with hypertension who had sought treatment regarding some allergic reaction. The published findings state that 9 of them were sensitive to sulfites. Why should we use the plus four method to compute a 95% confidence interval for the population proportion of sulfite sensitivity among patients with hypertension and an allergy?

A phycologist is interested in determining the proportion of algae samples from a local rivulet that belong to a particular phylum. A simple random sample of 50 alga is obtained and each alga is categorized as either being a cyanobacterium or not. The researcher found that 38 samples were, in fact, cyanobacteria. What is the standard error SE^p̂ of the proportion p̂ of the algae that were cyanobacteria?

A December 2013 Gallup poll of 13,186 nationally representative adults found that 8.9% reported being sick with a cold. In its report, Gallup warned that, "In addition to sampling error, question wording and practical difficulties in conducting surveys can introduce error or bias into the findings of public opinion polls." Which of the following sources of uncertainty is taken into account when computing the margin of error of the confidence interval?

A study examined a simple random sample of smokers who tried to quit smoking by using either a nicotine patch or e-cigarettes. Here is a quote from a news article referencing the study: "Quitting is hard. The number of people who quit (within 6 months) was low overall-just 38 of the 584 smokers given the e-cigarette or the patch." What is the margin of error for a plus four 95% confidence interval for the true population proportion of smokers who quit within 6 months using either e-cigarettes or the nicotine patch?

In 2013, Gallup surveyed a nationally representative sample of 2027 adults about their soda consumption. The survey showed that 24% mostly drank diet soda. This survey also found that a total of 56% drank mostly either regular or diet soda. Is this evidence that the majority of American adults drink mostly either regular or diet soda? To answer this question, you test the following hypotheses: H₀: p = 0.50, H_a: p > 0.50 What is the P-value for the test?

A study examined the effectiveness of Botox injections in the corrugator supercilii muscles for the treatment of chronic migraines. Of the 29 migraine-prone patients participating in the study, 24 reported some form of improvement. What is a plus four 95% confidence interval for the population proportion of migraine patients who benefit from such Botox injections? Assume the necessary assumptions for this confidence interval are met.

A phycologist is interested in determining the proportion of algae samples from a local rivulet that belong to a particular phylum. A simple random sample of 50 alga is obtained and each alga is categorized as either being a cyanobacterium or not. The researcher found that 38 samples were, in fact, cyanobacteria. What is the proportion p̂ of algae that were cyanobacteria?

In 2013, Gallup surveyed a nationally representative sample of 2027 adults about their soda consumption. The survey showed that 24% mostly drank diet soda. Based on the results, what is a 95% confidence interval for the proportion of American adults who mostly drink diet soda?